∫ k d x = k x + C {\displaystyle {\rm {\int kdx=kx+C}}} ∫ x a d x = 1 a + 1 x a + 1 + C {\displaystyle {\rm {\int x^{a}dx={\frac {1}{a+1}}x^{a+1}+C}}} ∫ e x d x = e x + C {\displaystyle {\rm {\int e^{x}dx=e^{x}+C}}} ∫ e a x d x = 1 a e a x + C {\displaystyle {\rm {\int e^{ax}dx={\frac {1}{a}}e^{ax}+C}}} ∫ a x d x = 1 ln a a x + C {\displaystyle {\rm {\int a^{x}dx={\frac {1}{\ln a}}a^{x}+C}}} ∫ 1 x d x = ln x + C {\displaystyle {\rm {\int {\frac {1}{x}}dx=\ln x+C}}} ∫ 1 a x + b d x = 1 a ln ( a x + b ) + C {\displaystyle {\rm {\int {\frac {1}{ax+b}}dx={\frac {1}{a}}\ln(ax+b)+C}}} ∫ sin x d x = − cos x + C {\displaystyle {\rm {\int \sin xdx=-\cos x+C}}} ∫ sin a x d x = − 1 a cos a x + C {\displaystyle {\rm {\int \sin axdx=-{\frac {1}{a}}\cos ax+C}}} ∫ cos x d x = sin x + C {\displaystyle {\rm {\int \cos xdx=\sin x+C}}} ∫ cos a x d x = 1 a sin a x + C {\displaystyle {\rm {\int \cos axdx={\frac {1}{a}}\sin ax+C}}} I n = ∫ 0 π 2 sin n x d x = ∫ 0 π 2 cos n x d x = n − 1 n I n − 2 {\displaystyle {\rm {I_{n}=\int _{0}^{\frac {\pi }{2}}\sin ^{n}xdx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}xdx={\frac {n-1}{n}}I_{n-2}}}} ∫ d x cos 2 x = ∫ sec 2 x d x = tan x + C {\displaystyle {\rm {\int {\frac {dx}{\cos ^{2}x}}=\int \sec ^{2}xdx=\tan x+C}}} ∫ d x sin 2 x = ∫ csc 2 x d x = − cot x + C {\displaystyle {\rm {\int {\frac {dx}{\sin ^{2}x}}=\int \csc ^{2}xdx=-\cot x+C}}} ∫ sec x ⋅ tan x d x = sec x + C {\displaystyle {\rm {\int \sec x\cdot \tan xdx=\sec x+C}}} ∫ csc x ⋅ cot x d x = − csc x + C {\displaystyle {\rm {\int \csc x\cdot \cot xdx=-\csc x+C}}} ∫ tan x d x = − ln | cos x | + C {\displaystyle {\rm {\int \tan xdx=-\ln |\cos x|+C}}} ∫ cot x d x = ln | sin x | + C {\displaystyle {\rm {\int \cot xdx=\ln |\sin x|+C}}} ∫ sec x d x = ln | sec ( x ) + tan ( x ) | + C {\displaystyle {\rm {\int \sec xdx=\ln |\sec(x)+\tan(x)|+C}}} ∫ csc x d x = ln | csc ( x ) − cot ( x ) | + C {\displaystyle {\rm {\int \csc xdx=\ln |\csc(x)-\cot(x)|+C}}} ∫ sinh x d x = cosh x + C {\displaystyle {\rm {\int \sinh xdx=\cosh x+C}}} ∫ cosh x d x = sinh x + C {\displaystyle {\rm {\int \cosh xdx=\sinh x+C}}} ∫ d x 1 − x 2 = arcsin x + C {\displaystyle {\rm {\int {\frac {dx}{\sqrt {1-x^{2}}}}=\arcsin x+C}}} ∫ d x a 2 − x 2 = arcsin x a + C {\displaystyle {\rm {\int {\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\arcsin {\frac {x}{a}}+C}}} ∫ d x 1 + x 2 = arctan x + C {\displaystyle {\rm {\int {\frac {dx}{1+x^{2}}}=\arctan x+C}}} ∫ d x a 2 + x 2 = 1 a arctan x a + C {\displaystyle {\rm {\int {\frac {dx}{a^{2}+x^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}+C}}} ∫ d x a 2 − x 2 = 1 2 a ln a + x a − x + C {\displaystyle {\rm {\int {\frac {dx}{a^{2}-x^{2}}}={\frac {1}{2a}}\ln {\frac {a+x}{a-x}}+C}}} ∫ d x x 2 − a 2 = 1 2 a ln | x − a x + a | + C {\displaystyle {\rm {\int {\frac {dx}{x^{2}-a^{2}}}={\frac {1}{2a}}\ln |{\frac {x-a}{x+a}}|+C}}} ∫ d x x 2 ± a 2 = ln ( x + x 2 ± a 2 ) + C {\displaystyle {\rm {\int {\frac {dx}{\sqrt {x^{2}\pm a^{2}}}}=\ln(x+{\sqrt {x^{2}\pm a^{2}}})+C}}} ∫ x 2 + a 2 d x = x 2 x 2 + a 2 + a 2 2 ln ( x + x 2 + a 2 ) + C {\displaystyle {\rm {\int {\sqrt {x^{2}+a^{2}}}dx={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}+{\frac {a^{2}}{2}}\ln(x+{\sqrt {x^{2}+a^{2}}})+C}}} ∫ x 2 − a 2 d x = x 2 x 2 − a 2 − a 2 2 ln | x + x 2 − a 2 | + C {\displaystyle {\rm {\int {\sqrt {x^{2}-a^{2}}}dx={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}-{\frac {a^{2}}{2}}\ln |x+{\sqrt {x^{2}-a^{2}}}|+C}}} ∫ a 2 − x 2 d x = x 2 a 2 − x 2 + a 2 2 arcsin x a + C {\displaystyle {\rm {\int {\sqrt {a^{2}-x^{2}}}dx={\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}+C}}}
